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A292612 a(n) = F(n)^2 + 4*(-1)^n = F(n+3)*F(n-3), where F = A000045.

%I A292612 #45 Feb 16 2025 08:33:51
%S A292612 4,-3,5,0,13,21,68,165,445,1152,3029,7917,20740,54285,142133,372096,
%T A292612 974173,2550405,6677060,17480757,45765229,119814912,313679525,
%U A292612 821223645,2149991428,5628750621,14736260453,38580030720,101003831725,264431464437,692290561604,1812440220357
%N A292612 a(n) = F(n)^2 + 4*(-1)^n = F(n+3)*F(n-3), where F = A000045.
%C A292612 This is the case k=3 of the identity F(n)^2 - F(k)^2*(-1)^(n+k) = F(n+k)*F(n-k), known also as Catalan's identity.
%H A292612 Colin Barker, <a href="/A292612/b292612.txt">Table of n, a(n) for n = 0..1000</a>
%H A292612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansIdentity.html">Catalan's Identity</a>.
%H A292612 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A292612 G.f.: (4 - 11*x + 3*x^2)/((1 + x)*(1 - 3*x + x^2)).
%F A292612 a(n) = a(-n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A292612 a(n) = 4*A001654(n+1) - 11*A001654(n) + 3*A001654(n-1) with A001654(-1)=0.
%F A292612 5*a(n) = Lucas(2*n) + 18*(-1)^n. Note that Lucas(2*n) + r*(-1)^n is divisible by 5 for r = -2, 3, -7, 8, -12, 13, -17, 18, -22, 23, -27, ... = (-1/4)*(3 + 5*(2*m+1)*(-1)^m) = (-1)^m*A047221(m). On the other hand, a(n) is divisible by 5 when n is a member of A047221.
%F A292612 a(n) = (1/5)*(18*(-1)^n + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n). - _Colin Barker_, Sep 20 2017
%F A292612 Sum_{n>=4} 1/a(n) = 143/960. - _Amiram Eldar_, Oct 05 2020
%F A292612 Sum_{n>=4} (-1)^n/a(n) = 3/(4*phi) - 407/960, where phi is the golden ratio (A001622). - _Amiram Eldar_, Oct 06 2020
%p A292612 with(combinat,fibonacci):  A292612:=seq(fibonacci(n)^2+4*(-1)^n, n=0..10^2); # _Muniru A Asiru_, Sep 26 2017
%t A292612 Table[Fibonacci[n]^2 + 4 (-1)^n, {n, 0, 40}]
%o A292612 (PARI) for(n=0, 40, print1(fibonacci(n)^2+4*(-1)^n", "));
%o A292612 (PARI) Vec((4-11*x+3*x^2)/((1+x)*(1-3*x+x^2))+O(x^30)) \\ _Colin Barker_, Sep 20 2017
%o A292612 (Sage) [fibonacci(n)^2+4*(-1)^n for n in range(40)]
%o A292612 (Magma) [Fibonacci(n)^2+4*(-1)^n: n in [0..40]];
%o A292612 (GAP)
%o A292612 List([0..10^2],n ->Fibonacci(n)^2+4*(-1)^n); # _Muniru A Asiru_, Sep 26 2017
%Y A292612 Cf. A000045, A001622, A005248: Lucas(2*n), A001654: F(n)*F(n+1).
%Y A292612 Cf. A007598 (k=0), A059929 (k=1, without initial 1), A192883 (k=2, without initial -1), this sequence (k=3).
%K A292612 sign,easy
%O A292612 0,1
%A A292612 _Bruno Berselli_, Sep 20 2017